Question

If $f(x) = 2{x^6} + 3{x^4} + 4{x^2}$ then $f'(x)$ is:
A. Even function
B. An odd function
C. Neither even nor odd
D. None of these

Answer 1

Here the given question is to find the relation between the expression, that is the expression is either even or odd function. Here we know the formulae which satisfies even function and for odd function also, if the expression satisfies accordingly then the solution will be accordingly.
Formulae Used: For even function, for any function say $f(x)$ then:
$\Rightarrow f(x) - f( - x) = 0$
For odd function, for any function say $f(x)$ then:
$\Rightarrow f(x) + f( - x) = 0$

Complete step by step answer:
Here in the above question we are provided with an expression and need to find its validity as an even or odd function, and here we need to solve accordingly. The question requires first the derivative of the function and then we need to check for its validity as even or odd function, on solving for derivative we get:

$$\Rightarrow f'(x) = \dfrac{d}{{dx}}\left( {2{x^6} + 3{x^4} + 4{x^2}} \right) \\\ \Rightarrow f'(x) = 6 \times 2{x^{6 - 1}} + 4 \times 3{x^{4 - 1}} + 2 \times 4{x^{2 - 1}} \\\ \Rightarrow f'(x) = 12{x^5} + 12{x^3} + 8x \\\$$

Now we need to solve according to the formulae of even and odd function, on solving we have:
For even function:

\Rightarrow f(x) - f( - x) = 0 \\\ \Rightarrow \left( {12{x^5} + 12{x^3} + 8x} \right) - \left\\{ { - \left( {12{x^5} + 12{x^3} + 8x} \right)} \right\\} = 0 \\\ \Rightarrow 12{x^5} + 12{x^3} + 8x + 12{x^5} + 12{x^3} + 8x = 0 \\\ \Rightarrow 24{x^5} + 12{x^3} + 16x = 0 \\\

Here we can see that the expression is not satisfying for even function, as both sides of the equation are not equal.
Now checking for odd function we have:

\Rightarrow f(x) + f( - x) = 0 \\\ \Rightarrow \left( {12{x^5} + 12{x^3} + 8x} \right) + \left\\{ { - \left( {12{x^5} + 12{x^3} + 8x} \right)} \right\\} = 0 \\\ \Rightarrow 12{x^5} + 12{x^3} + 8x - 12{x^5} - 12{x^3} - 8x = 0 \\\ \Rightarrow 0 = 0 \\\

Here we can see that both sides of the equation have the same value as zero.

So, the correct answer is “Option B”.

Note: For such questions which need to satisfy the expression for any particular property, here we need to solve for the related formulae by putting the value associated in the expression and then solve further, and check that the equation satisfies the formulae of not.